Estimation, Evaluation, and Selection of Actuarial Models

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60 CHAPTER 3. SAMPLING PROPERTIES OF ESTIMATORS From Example 3.30 we have ˆµ =6.1379 and ˆσ =1.3894 and an estimated covariance matrix of · ¸ 0.0965 0 ˆΣ/n = . 0 0.0483 The function is g(µ, σ) =exp(µ + σ 2 /2). The partial derivatives are ∂g ∂µ = exp(µ + σ2 /2) ∂g ∂σ = σ exp(µ + σ2 /2) and the estimates of these quantities are 1,215.75 and 1,689.16 respectively. The delta method produces the following approximation: dVar[g(ˆµ, ˆσ)] = £ 1, 215.75 1, 689.16 ¤ · 0.0965 0 0 0.0483 ¸· 1, 215.75 1, 689.16 ¸ = 280, 444. The confidence interval is 1, 215.75 ± 1.96 √ 280, 444 or 1, 215.75 ± 1, 037.96. The customary confidence interval for a population mean is ¯x ± 1.96s/ √ n. For Data Set B the interval is 1, 424.4 ± 1.96(3, 435.04)/ √ 20 or 1, 424.4 ± 1, 505.47. It is not surprising that this is a wider interval because we know that (for a lognormal population) the maximum likelihood estimator is asymptotically UMVUE. ¤ Exercise 69 Use the delta method to construct a 95% confidence interval for the mean of a gamma distribution using Data Set B. Preliminary calculations are in Exercise 67. Exercise 70 (*) For a lognormal distribution with parameters µ and σ you are given that the maximum likelihood estimates are ˆµ =4.215 and ˆσ =1.093. The estimated covariance matrix of (ˆµ, ˆσ) is · 0.1195 0 0 0.0597 The mean of a lognormal distribution is given by exp(µ + σ 2 /2). Estimate the variance of the maximum likelihood estimator of the mean of this lognormal distribution, using the delta method. Exercise 71 (*) A distribution has two parameters, α and β. A sample of size 10 produced the following loglikelihood function: ¸ . l(α, β) =−2.5α 2 − 3αβ − β 2 +50α +2β + k where k is a constant. Estimate the covariance matrix of the maximum likelihood estimator (ˆα, ˆβ). Exercise 72 In Exercise 46 two maximum likelihood estimates were obtained for the same model. The second estimate was based on more information than the first one. It would be reasonable to expectthatthesecondestimateismoreaccurate. Confirm this by estimating the variance of each of the two estimators. Do your calculations using the observed likelihood.
Chapter 4 Model evaluation and selection 4.1 Introduction When using data to build a model, the process must end with the announcement of a “winner.” While qualifications, limitations, caveats, and other attempts to escape full responsibility are appropriate, and often necessary, a commitment to a solution is required. In this Chapter we look at a variety of ways to evaluate a model and compare competing models. But we must also remember that whatever model we select, it is only an approximation of reality. This is reflected in the following modeler’s motto: 1 “All models are wrong, but some models are useful.” Thus, the goal of this Chapter is to determine a model that is good enough to use to answer the question. The challenge here is that the definition of “good enough” will depend on the particular application. Another important modeling point is that a solid understanding of the question will guide you to the answer. The following quote from John Tukey sums this up: “Far better an approximate answer to the right question, which is often vague, than an exact answer to the wrong question, which can always be made precise.” In this Chapter, a specific modeling strategy will be considered. For Part 4, our preference is to have a single approach that can be used for any probabilistic modeling situation. A consequence is that for any particular modeling situation there may be a better (more reliable or more accurate) approach. For example, while maximum likelihood is a good estimation method for most settings, it may not be the best 2 for certain distributions. A literature search will turn up methods that have been optimized for specific distributions, but they will not be mentioned here. Similarly, many of the hypothesis tests used here give approximate results. For specific cases, better approximations, or maybe even exact results, are available. They will also be bypassed. The goal here is to outline a method that will give reasonable answers most of the time and be adaptable to a variety of situations. 1 It is usually attributed to George Box. 2 There are many definitions of “best.” Combining the Cramér-Rao lower bound with Theorem 3.29 indicates that maximum likelihood estimators are asymptotically optimal using unbiasedness and minimum variance as the definition of best. 61
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